Abstract

Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant (Yu and Wang, 2009). Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009).

1. Introduction

While in general an interior region-of-interest (ROI) cannot be uniquely reconstructed from projection data only associated with lines through the ROI [1, 2], in the compressed sensing framework, we recently found that a two-dimensional interior ROI can be exactly reconstructed via the total variation minimization if the function on the ROI is piecewise constant [3, 4]. The major idea behind our analysis is that the total variations of a piecewise constant function and a smooth artifact function are separable. The main mathematical tool is the expression of the two-dimensional gradient in terms of the Dirac delta function. In our analysis [3], the Delta function was instrumental but applied heuristically without mathematical rigor. In this note, we will prove rigorously a more general theorem, as an extension of the total variation minimization property presented in [3], to characterize the total variation minimization property for a piecewise constant function defined on a domain in any dimension. Such a theorem may serve as a theoretical basis for further development of interior tomography algorithms.

2. Theoretical Result

For piecewise constant or piecewise smooth functions, it is natural to use the space of functions of bounded variation [5] to capture the discontinuities. For an integer , let be a -dimensional open bounded set and denote its boundary by . Then the space of functions of bounded variation is
It is a Banach space with the norm
where is the integral of over , and
is the total variation of the function . Here is the space of continuously differentiable functions that vanish on , and for , . The Sobolev space
is a subspace of and

We that assume has a piecewise boundary, and it is decomposed into a union of a finite number of subsets with disjoint interiors
such that each subset has a piecewise boundary. The unit outward normal vector on is denoted by . Denote , which may be empty for some pairs of and between 1 and . We write for the ()-dimensional measure of ; it is the area of for , and the length of for . The symbol will refer to a summation for those and with a nonempty in the range . The main result of this note is the following.

Theorem 1. Let be a piecewise constant function corresponding to the decomposition (6): for , . Then we have
Consequently, we have the minimization property

Proof. After an integration by parts and some rearrangement, we have, for any ,
By the definition (3),
Taking in (11), the formula (8) follows (cf. the argument in the next paragraph for a more general situation). Moreover, from (11) again,
For the opposite inequality, we first consider the case where . For any , define two open subsets
Here, is the distance between and a closed set . Obviously, for some constant ,
We start with a function satisfying
and then apply the well-known mollification technique in the theory of Sobolev space [6] to define
where is the ball of radius centered at the origin, , and
Then for , and for sufficiently small, . Moreover, as , converges uniformly to for . Thus from (11),
and as , we obtain, with some constant
Using the defining properties of , we further have
for some other constant . Since is arbitrary, we obtain from the above relation that
Combining (12) and (21), we conclude (7) for .For , we use the density of in [6] and choose such that
Since (3) defines a seminorm on , we have
Thus, taking this limit in (7) for , we obtain (7) for .

As an example of (8), let be a disk of radius centered at the origin. Consider a piecewise constant, radial function defined on such that it has a jump at , , where . Then by (8), we have
(cf. [3, Theorem ]).

3. Discussions and Conclusion

Some comments on the name of our approach “CS-based interior tomography” are in order. In the strict sense, compressed sensing refers to situations where the sampling scheme is built (often with random techniques) to achieve specific properties for satisfactory recovery of an underlying signal, rather than imposed by a specific detector arrangement as in limited data tomography. However, in a broad sense, compressed sensing can be interpreted as achieving better reconstruction from less data relative to the common practice. Hence, while the current name is not far off, an alternative phrase for our approach can be “total variation minimization-based interior tomography.”

In conclusion, we have extended the total variation minimization property of a piecewise function from two-dimensions to any dimensionality in the Sobolev space, which can be used for exact reconstruction of any piecewise function on an ROI by minimizing its total variation under the constraint of the truncated projection data through the ROI. Previously, we implemented an alternating iterative reconstruction algorithm to minimize the total variation, which is time-consuming and needs improvement. Under the guidance of the theoretical finding presented here, we are working to develop a multidimensional ROI reconstruction algorithm for better performance. Clearly, major efforts are still needed in this direction.

Acknowledgments

This work is partially supported by NIH/NIBIB Grants (EB002667, EB004287, EB007288) and a Grant from Toshiba Medical Research Institute USA, Inc.